Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors
- Cambridge: Cambridge University Press, 2001.
- xvii, 461p.: pbk.: 23cm.
- Cambridge Texts in Applied Mathematics .
Includes References and Index
This book develops the theory of global attractors for a class of parabolic PDEs which includes reaction-diffusion equations and the Navier-Stokes equations, two examples that are treated in detail. A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment of existence and uniqueness of solutions for both linear time-independent problems (Poisson's equation) and the nonlinear evolution equations which generate the infinite-dimensional dynamical systems of the title. Attention then switches to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space which determines the asymptotic dynamics. In particular, the concluding chapters investigate in what sense the dynamics restricted to the attractor are themselves 'finite-dimensional'. The book is intended as a didactic text for first year graduates, and assumes only a basic knowledge of Banach and Hilbert spaces, and a working understanding of the Lebesgue integral.
Develops theory of PDEs as dynamical systems, theory of global attractors, and some consequences of that theory Only a low level of previous knowledge of functional analysis is assumed, so accessible to the widest possible mathematical audience Numerous exercises, with full solutions available on the web